Bivariate Statistics - IB Math AA Question Bank

Question 1

SL • Paper 2

Easy

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SL • Paper 2

Easy

Calculator Permitted

Five paired observations of variables $x$ and $y$ are shown in the scatter diagram. A student draws the line $y=2.8x+14$ as a proposed line of best fit. The coordinates of the five observations, in order of increasing $x$ , are $(3,22)$ , $(5,29)$ , $(6,31)$ , $(10,42)$ and $(11,48)$ .

A

Find the coordinates of the mean point, $(\bar{x},\bar{y})$ .

[2]

B

Determine whether the student's line is suitable as a line of best fit by eye. Give a reason for your answer.

[2]

Question 2

SL • Paper 1

Medium

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SL • Paper 1

Medium

Non Calculator

For a sample of students, $x$ denotes the number of hours spent revising and $y$ denotes the score on a test. The regression line of $y$ on $x$ is

y=5x+42

for $1\leq x\leq 8$ . The Pearson product-moment correlation coefficient is $r=0.92$ . For this sample size, the critical value for testing linear correlation is $0.754$ .

A

State the direction and strength of the linear correlation.

[1]

B

Determine whether the linear correlation is significant, giving a reason.

[2]

C

Use the regression line to estimate the test score for a student who revised for $6$ hours.

[2]

D

Explain why using this regression line to estimate the score for a student who revised for $12$ hours may be unreliable.

[1]

Question 3

SL • Paper 1

Medium

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SL • Paper 1

Medium

Non Calculator

The height, $h$ cm, of a young plant $t$ weeks after it is planted is modelled by the regression line

h=4t+12

for $2\leq t\leq 10$ .

A

Interpret the meaning of the gradient in this context.

[1]

B

Interpret the meaning of the intercept, and comment on its validity in this context.

[2]

C

Find the predicted height of the plant after $7$ weeks.

[2]

D

Find the predicted increase in height over any $3$ week interval.

[1]

Question 4

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Five bivariate observations have variables $x$ and $y$ . For these observations,

\sum x=30,\qquad \sum y=50

A line of best fit drawn by eye passes through the mean point and also through the point $(2,4)$ .

A

Find the coordinates of the mean point.

[2]

B

Find the equation of this line of best fit in the form $y=mx+c$ .

[3]

C

Use your line to estimate $y$ when $x=8$ .

[1]

Question 5

SL • Paper 1

Medium

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SL • Paper 1

Medium

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For a set of bivariate data, the regression line of $y$ on $x$ is

y=-2x+37

The observed values of $x$ satisfy $5\leq x\leq 13$ .

A

State the direction of the linear correlation suggested by the regression line.

[1]

B

Use the regression line to estimate $y$ when $x=9$ .

[2]

C

Find the predicted change in $y$ when $x$ increases by $4$ .

[2]

D

Classify a prediction made using $x=4$ and a prediction made using $x=12$ as interpolation or extrapolation.

[2]

Question 6

HL • Paper 1

Medium

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HL • Paper 1

Medium

Non Calculator

For a set of bivariate data, the mean point is $(4,15)$ . The regression line of $x$ on $y$ has gradient $\dfrac{2}{5}$ .

A

Determine the equation of the regression line of $x$ on $y$ .

[2]

B

Use your equation to predict $x$ when $y=20$ .

[2]

C

The regression line of $y$ on $x$ is $y=\dfrac{5}{4}x+10$ . Verify that this line also passes through the mean point.

[1]

Question 7

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A juice bar recorded the daily maximum temperature, $x^\circ\text{C}$ , and the number of cold drinks sold, $y$ , on six different days. The data are shown in the table.

Maximum temperature [°C]	Cold drinks sold
12	78
14	81
16	92
18	101
20	107
22	117

A

Find Pearson's product-moment correlation coefficient, $r$ , for the data.

[2]

B

The regression line of $y$ on $x$ has equation $y=ax+b$ . Find the value of $a$ and the value of $b$ .

[2]

C

Use the regression line to estimate the number of cold drinks sold on a day when the maximum temperature is $19^\circ\text{C}$ .

[2]

Question 8

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A teacher recorded the number of absences, $x$ , and the final test score, $y$ , for six students in a class. The data are shown in the table.

	1	2	3	4	5	6
number of absences, x	1	2	3	4	5	6
final test score, y	75	74	68	62	56	52

A

Find Pearson's product-moment correlation coefficient, $r$ .

[2]

B

Find the equation of the regression line of $y$ on $x$ .

[2]

C

State what the gradient means in this context.

[1]

Question 9

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

An athletics coach records the number of training sessions per week, $x$ , and the distance run in a fitness test, $y$ km, for six athletes. The data are shown in the table. For $n=6$ , the critical value of $|r|$ at the $5\%$ significance level is $0.811$ .

Variable	Athlete 1	Athlete 2	Athlete 3	Athlete 4	Athlete 5	Athlete 6
training sessions per week, x	1	2	3	4	5	6
distance run, y [km]	1	3	4	7	9	10

A

Find Pearson's product-moment correlation coefficient, $r$ .

[2]

B

Determine whether there is significant positive linear correlation at the $5\%$ level.

[2]

C

The coach claims that increasing the number of training sessions causes athletes to run further in the test. Comment on this claim.

[1]

Question 10

SL • Paper 1

Medium

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SL • Paper 1

Medium

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In an investigation, $x$ represents temperature and $y$ represents the time taken for a reaction. The regression line of $y$ on $x$ passes through the mean point $(20,74)$ and has gradient $-\dfrac{3}{2}$ .

A

Find the equation of the regression line in the form $y=ax+b$ .

[2]

B

Use the regression line to estimate $y$ when $x=16$ .

[2]

C

For one observation, $x=16$ and the actual value of $y$ was $83$ . Find the difference actual value minus predicted value.

[1]

D

Explain why this regression line alone does not prove that temperature causes a change in reaction time.

[1]

Question 11

SL • Paper 1

Medium

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SL • Paper 1

Medium

Non Calculator

A set of bivariate data has Pearson product-moment correlation coefficient $r=-0.08$ . The observed values of $x$ satisfy $-3\leq x\leq 3$ . A scatter diagram of the data shows points lying close to a curve shaped like a parabola.

A

State what the value of $r$ suggests about the linear correlation.

[1]

B

Explain why the value of $r$ may be misleading if used on its own.

[2]

C

State whether a linear regression model would be appropriate for these data, giving a reason.

[2]

D

Explain why a prediction made using $x=5$ would be particularly unreliable.

[1]

Question 12

HL • Paper 1

Medium

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HL • Paper 1

Medium

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For a set of bivariate data, the regression line of $y$ on $x$ is

y=\frac{1}{2}x+4

and the regression line of $x$ on $y$ is

x=\frac{3}{2}y-3

A

Find the mean point of the data.

[3]

B

Use the appropriate regression line to estimate $x$ when $y=14$ .

[2]

C

Explain why rearranging the regression line of $y$ on $x$ is not the appropriate method for part (b).

[1]

Question 13

HL • Paper 1

Medium

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HL • Paper 1

Medium

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For a set of data, the regression line of $x$ on $y$ is

x=\frac{2}{3}y+5

The regression line of $y$ on $x$ is

y=\frac{3}{4}x+6

The observed values of $y$ satisfy $12\leq y\leq 30$ .

A

Use the appropriate regression line to estimate $x$ when $y=21$ .

[2]

B

State whether using the regression line of $x$ on $y$ when $y=34$ would be interpolation or extrapolation.

[1]

C

Use the appropriate regression line to estimate $y$ when $x=23$ .

[2]

D

Explain why the equation $x=\dfrac{2}{3}y+5$ should not be rearranged to answer part (c).

[1]

Question 14

HL • Paper 1

Medium

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HL • Paper 1

Medium

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For a set of bivariate data, the regression line of $y$ on $x$ is

y=\frac{1}{2}x+2

and the regression line of $x$ on $y$ is

x=\frac{4}{3}y

A

Find the mean point of the data.

[3]

B

Use the appropriate regression line to estimate $y$ when $x=10$ .

[2]

C

Use the appropriate regression line to estimate $x$ when $y=9$ .

[1]

Question 15

HL • Paper 1

Medium

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HL • Paper 1

Medium

Non Calculator

For a data set, the regression line of $y$ on $x$ is

y=-2x+22

and the regression line of $x$ on $y$ is

x=-\frac{1}{8}y+\frac{29}{4}

A

Find the mean point of the data.

[3]

B

Use the appropriate regression line to estimate $x$ when $y=18$ .

[2]

C

Interpret the gradient of the regression line of $x$ on $y$ .

[1]

Question 16

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A scientist models the relationship between a variable $x$ and a positive variable $y$ . The table gives values of $x$ and $\ln y$ . The regression equation of $\ln y$ on $x$ is written as $\ln y=ax+b$ .

Observation	1	2	3	4	5
x	2	4	6	8	10
ln y	1.182	1.647	2.112	2.577	3.042

A

Find the value of $a$ and the value of $b$ .

[2]

B

Use the regression equation to estimate $y$ when $x=7$ .

[3]

C

State whether this estimate is an interpolation or an extrapolation.

[1]

Question 17

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A taxi company records the distance of a journey, $x$ km, and the fare, $y$ dollars, for five journeys. The data are shown in the table.

Journey	1	2	3	4	5
distance $x$ [km]	4.491	5.921	7.600	9.779	12.209
fare $y$ [dollars]	9	13	17	21	25

A

Find Pearson's product-moment correlation coefficient, $r$ .

[2]

B

Find the regression line of $x$ on $y$ , in the form $x=cy+d$ .

[2]

C

Use your regression line to estimate the distance of a journey with fare $20$ dollars.

[2]

Question 18

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A sports scientist records the age, $x$ , in decades, and sprint speed, $y$ , in $\text{km}\,\text{h}^{-1}$ , of five athletes. The data are shown in the table.

Athlete	1	2	3	4	5
Age / decades	3	4	5	6	7
Sprint speed / $\text{km}\,\text{h}^{-1}$	40.91	34.39	34.20	29.79	31.71

A

Find the regression line of $y$ on $x$ .

[2]

B

Find the regression line of $x$ on $y$ .

[2]

C

Estimate the age, in decades, of an athlete whose sprint speed is $33$ $\text{km}\,\text{h}^{-1}$ . State which regression line you used.

[2]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

For a set of bivariate data, the regression line of $y$ on $x$ is $y=1.84x+12.5$ and Pearson's product-moment correlation coefficient is $r=0.912$ . New variables $u$ and $v$ are defined by $u=x-5$ and $v=2y+10$ .

A

Determine Pearson's product-moment correlation coefficient between $u$ and $v$ .

[1]

B

Find the regression line of $v$ on $u$ .

[3]

Question 20

SL • Paper 1

Medium

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SL • Paper 1

Medium

Non Calculator

A small museum recorded the number of days, $x$ , after the opening of an exhibition and the number of visitors, $y$ , on that day. A linear regression model for the data is

y=-35x+820,\qquad 1\leq x\leq 14

The mean of the recorded $x$ -values is $8$ , and there were $10$ recorded days.

A

I.

Find the mean number of visitors per recorded day.

[2]

II.

Hence find the total number of visitors on the recorded days.

[1]

B

I.

Use the regression model to estimate the number of visitors on day $5$ .

[2]

II.

Find the predicted change in the number of visitors over any $4$ -day interval.

[2]

C

I.

On day $5$ the actual number of visitors was $630$ . Find the residual, defined as actual value minus predicted value.

[2]

II.

State whether using the model to estimate the number of visitors on day $20$ would be interpolation or extrapolation.

[1]

D

The museum manager claims that each extra day after opening causes exactly $35$ fewer visitors. Comment on this claim.

[2]

Question 21

SL • Paper 1

Medium

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SL • Paper 1

Medium

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A shop selling second-hand cameras records the age, $x$ years, and the repair cost, $y$ dollars, for six cameras. For these data,

\sum x=48,\qquad \sum y=150

A line of best fit drawn by eye must pass through the mean point. A student suggests the line $y=3x+1$ .

A

I.

Find the coordinates of the mean point.

[2]

II.

Determine whether the student's line satisfies the mean point condition.

[2]

B

I.

Use the student's line to estimate the repair cost for a camera aged $10$ years.

[1]

II.

The regression line of $y$ on $x$ for these data is $y=\frac{5}{2}x+5$ , for $4\leq x\leq 12$ . Interpret the gradient in context.

[2]

C

Use the regression line $y=\frac{5}{2}x+5$ to estimate the repair cost for a camera aged $14$ years, and comment on the reliability of this estimate.

[3]

Question 22

SL • Paper 1

Medium

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SL • Paper 1

Medium

Non Calculator

A city council records the floor area, $x$ , in hundreds of square metres, and the annual number of visits, $y$ , in thousands, for several community libraries. For these data, Pearson's product-moment correlation coefficient is $r=0.84$ . For this sample size, the critical value for testing linear correlation is $0.754$ .

The regression line of $y$ on $x$ is

y=2x+18,\qquad 20\leq x\leq 60

A

I.

(a)(i) Describe the direction and strength of the linear correlation.

[2]

II.

(a)(ii) Determine whether the linear correlation is significant, giving a reason.

[2]

B

I.

(b)(i) Estimate the annual number of visits for a library with floor area $45$ hundreds of square metres.

[2]

II.

(b)(ii) Interpret the intercept of the regression line, and comment on its validity in this context.

[2]

C

I.

(c)(i) A councillor claims that increasing the floor area causes an increase in the number of visits. Comment on this claim.

[2]

II.

(c)(ii) Explain why the regression line should not automatically be rearranged to estimate $x$ from a given value of $y$ .

[1]

Question 23

SL • Paper 1

Medium

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SL • Paper 1

Medium

Non Calculator

For a sample of adults, $x$ denotes the number of phone notifications received in one hour and $y$ denotes the number of minutes of uninterrupted reading in that hour. The regression line of $y$ on $x$ has gradient $-\frac{3}{2}$ and passes through the mean point $(12,54)$ . The observed values of $x$ satisfy $4\leq x\leq 18$ .

A

Find the equation of the regression line of $y$ on $x$ , in the form $y=ax+b$ .

[3]

B

Interpret the regression line in the following parts.

I.

Interpret the gradient in context.

[2]

II.

Interpret the intercept, and comment on whether this interpretation is reliable.

[2]

C

Use the regression line to answer the following parts.

I.

Estimate the uninterrupted reading time for an adult receiving $16$ notifications in one hour.

[2]

II.

For one adult with $16$ notifications, the actual uninterrupted reading time was $45$ minutes. Find the residual, actual value minus predicted value. If you did not obtain an estimate in part (c)(i), use $48$ minutes.

[2]

D

Explain why the regression equation alone does not prove that phone notifications cause reduced uninterrupted reading time.

[2]

Question 24

SL • Paper 1

Medium

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SL • Paper 1

Medium

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A craft workshop records the drying time, $x$ hours, and the percentage moisture remaining, $y$ , for a set of clay tiles. A regression model for predicting $y$ from $x$ is

y=-4x+86,\qquad 2\leq x\leq 10

For these data, $r=-0.96$ . For this sample size, the critical value for testing linear correlation is $0.878$ .

A

I.

State the variable that should be used as the predictor in this regression model.

[1]

II.

State the variable being predicted by this regression model.

[1]

B

I.

Use the model to estimate the percentage moisture remaining after $6$ hours.

[2]

II.

A student rearranges the model to estimate the drying time when $50$ percent moisture remains. Find the student's estimate, and state one limitation of this method.

[3]

C

I.

Determine whether the linear correlation is significant, giving a reason.

[2]

II.

Explain why using the model at $x=14$ hours may be unreliable.

[1]

Question 25

HL • Paper 1

Medium

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HL • Paper 1

Medium

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For a set of bivariate data, the regression line of $x$ on $y$ is

x=\frac{1}{3}y+7

The mean point of the data is $(10,9)$ . The regression line of $y$ on $x$ has gradient $2$ . A student rearranges the regression line of $x$ on $y$ to obtain $y=3x-21$ and uses this to predict $y$ from $x$ .

A

Use the regression line of $x$ on $y$ to predict $x$ when $y=15$ .

[2]

B

Find the equation of the regression line of $y$ on $x$ .

[2]

C

For $x=12$ , find the prediction from the regression line of $y$ on $x$ and the prediction from the student's rearranged equation.

[2]

D

State which prediction in part (c) should be used, giving a reason.

[1]

Question 26

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A bicycle hire company records the number of sunshine hours, $x$ , and the number of bicycles hired, $y$ , on seven Saturdays. The paired data are
$(2,18),(4,35),(5,39),(7,54),(8,63),(10,76),(11,82)$

A

I.

Find Pearson's product-moment correlation coefficient, $r$ .

[2]

II.

Comment on the strength and direction of the linear correlation.

[1]

B

I.

Find the equation of the regression line of $y$ on $x$ in the form $y=ax+b$ .

[2]

II.

Use the regression line to estimate the number of bicycles hired on a Saturday with $9$ sunshine hours.

[2]

C

A manager wants to use the regression line to estimate the number of bicycles hired on a Saturday with $15$ sunshine hours. State whether this would be interpolation or extrapolation, and comment on its reliability.

[1]

Question 27

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

An events organiser records the rainfall, $x$ mm, and the attendance, $y$ , at seven outdoor performances. The paired data are
$(0,510),(2,480),(5,430),(7,410),(9,360),(12,325),(14,290)$
For $n=7$ , the critical value of $|r|$ at the $5\%$ significance level is $0.754$ .

A

I.

Find Pearson's product-moment correlation coefficient, $r$ .

[2]

II.

Determine whether there is significant linear correlation at the $5\%$ level.

[2]

B

I.

Find the regression line of $y$ on $x$ .

[2]

II.

Estimate the attendance when the rainfall is $6$ mm.

[1]

C

I.

State whether the estimate in part (b)(ii) is interpolation or extrapolation.

[1]

II.

The organiser claims that rainfall causes lower attendance. Comment on this claim.

[1]

Question 28

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A city engineer records the traffic flow, $x$ , in hundreds of vehicles per hour, and the noise level, $y$ dB, at six roads. The paired data are
$(12,55),(18,59),(21,61),(27,66),(33,71),(39,74)$

A

I.

Find the coordinates of the mean point $(\bar{x},\bar{y})$ .

[2]

II.

Explain why a line of best fit drawn by eye should pass close to this point.

[1]

B

I.

Find the regression line of $y$ on $x$ .

[2]

II.

Use the regression line to estimate the noise level when $x=30$ .

[1]

C

At another road, the traffic flow was $x=30$ and the actual noise level was $70$ dB. Find the residual, using actual value minus predicted value, and interpret it.

[2]

Question 29

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

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Calculator Permitted

The table shows five paired observations of variables $x$ and $y$ .

$x$	$y$
2.4	6.971
4.4	9.693
5.4	12.600
6.4	15.507
8.4	18.229

A

Find the regression line of $y$ on $x$ .

[2]

B

Find the regression line of $x$ on $y$ .

[2]

C

The two regression lines intersect at the mean point of the data. Determine the coordinates of this point.

[2]

Question 30

HL • Paper 2

Medium

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HL • Paper 2

Medium

Calculator Permitted

A marine biologist records the length, $x$ cm, and mass, $y$ g, of six fish. One observation appears to be unusual. The data are shown in the table.

Observation	1	2	3	4	5	6
Length x [cm]	1	2	3	4	5	6
Mass y [g]	2.00	4.23	6.10	8.21	9.96	3.00

A

Find Pearson's product-moment correlation coefficient, $r$ , using all six observations.

[2]

B

The biologist removes the unusual observation and recalculates the correlation coefficient using the remaining five observations. Find the new value of $r$ .

[2]

C

Using only the remaining five observations, the regression line of $y$ on $x$ is $y=1.99x+0.13$ . Estimate the mass of a fish of length $6$ cm.

[2]

D

Comment on the reliability of the estimate in part (c).

[1]

Question 31

HL • Paper 2

Medium

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HL • Paper 2

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Calculator Permitted

The population $P$ of a bacterial culture, in thousands, is recorded at time $t$ hours. The table gives values of $t$ and $\ln P$ . It is suggested that $P$ may be modelled by $P=Ae^{kt}$ , where $A$ and $k$ are constants.

t [h]	ln P
0	1.108
2	1.472
4	1.836
6	2.200
8	2.564

A

Find the regression equation of $\ln P$ on $t$ in the form $\ln P=mt+c$ .

[2]

B

Hence find the value of $A$ and the value of $k$ .

[2]

C

Use the model to estimate the time at which the population reaches $20$ thousand.

[3]

Question 32

HL • Paper 2

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HL • Paper 2

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The table shows values of $x$ and $y$ for seven observations. The table suggests a curved relationship.

$x$	$y$
-3	9.18
-2	4.14
-1	1.10
0	0.08
1	1.06
2	4.06
3	9.08

A

Find Pearson's product-moment correlation coefficient, $r$ , for $x$ and $y$ .

[2]

B

Explain why the value of $r$ may be misleading in this context.

[1]

C

Let $z=x^2$ . Find the regression line of $y$ on $z$ .

[2]

D

Use the regression line in part (c) to estimate $y$ when $x=2.5$ .

[2]

Question 33

HL • Paper 3

Medium

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HL • Paper 3

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A museum records the age $x$ years of ceramic samples and their measured hardness $y$ on a standard scale. The regression line of $x$ on $y$ has gradient $0.03125$ and passes through the mean point. The mean age is $8.00$ years and the Pearson correlation coefficient is $0.750$ .

A

I.

Find the mean hardness of the samples if the regression line of $x$ on $y$ is $x=0.03125y+2.50$ .

[2]

II.

Find the equation of the regression line of $y$ on $x$ .

[2]

B

I.

Estimate the hardness of a sample aged $7.20$ years. If you did not obtain an equation in part (a)(ii), use $y=18.0x+32.0$ .

[2]

II.

Estimate the age of a sample with hardness $170$ .

[1]

C

I.

Find the ratio $s_y:s_x$ of the sample standard deviations.

[2]

II.

Explain why the intercept $32.0$ should be interpreted with care.

[2]

Question 34

SL • Paper 1

Hard

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SL • Paper 1

Hard

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A harbour officer records the time, $x$ hours after midnight, and the depth of water, $y$ metres, at six times during one tide cycle. The observations are

(0,9),\ (1,5),\ (2,3),\ (3,3),\ (4,5),\ (5,9)

A

I.

Find $\bar x$ .

[1]

II.

Find $\bar y$ .

[1]

B

For these data, $\sum (x-\bar x)^2>0$ and $\sum (y-\bar y)^2>0$ . Calculate Pearson's product-moment correlation coefficient, $r$ .

[3]

C

I.

State what the value of $r$ suggests about the linear correlation.

[1]

II.

Explain why using only the value of $r$ may be misleading for these data.

[2]

III.

State whether a linear regression model is appropriate for these data, giving a reason.

[1]

Question 35

HL • Paper 1

Hard

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HL • Paper 1

Hard

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For a set of bivariate data, the regression line of $y$ on $x$ has equation

y=\frac{1}{3}x+p

The regression line of $x$ on $y$ has equation

x=\frac{3}{4}y+6

The mean of the $x$ -values is $12$ .

A

I.

Find the value of $p$ .

[3]

II.

Write down the mean point.

[2]

B

I.

Use the appropriate regression line to estimate $y$ when $x=18$ . If you did not obtain $p=4$ , use $p=4$ .

[2]

II.

Use the appropriate regression line to estimate $x$ when $y=16$ .

[2]

C

A student rearranges $x=\frac{3}{4}y+6$ and uses it to predict $y$ when $x=18$ . Compare this prediction with the prediction from part (b)(i), and state which should be used.

[3]

Question 36

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A courier company records the distance of a delivery, $x$ km, and the delivery time, $y$ minutes. The regression line of $y$ on $x$ is

y=2x+10

The regression line of $x$ on $y$ has gradient $\frac{1}{5}$ . The mean point of the data is $(20,50)$ . The observed range of $x$ is $10\leq x\leq 28$ and the observed range of $y$ is $30\leq y\leq 80$ ; in each case, the endpoints are the observed minimum and maximum values.

A

I.

Find the equation of the regression line of $x$ on $y$ .

[3]

II.

Verify that the regression line of $y$ on $x$ passes through the mean point.

[1]

B

I.

Use the appropriate regression line to estimate the delivery time for a distance of $25$ km.

[2]

II.

Use the appropriate regression line to estimate the distance for a delivery time of $70$ minutes. If you did not obtain an equation in part (a)(i), use $x=\frac{1}{5}y+10$ .

[2]

III.

Explain why rearranging $y=2x+10$ is not the appropriate method for part (b)(ii).

[2]

C

I.

Classify the two predictions in part (b) as interpolation or extrapolation.

[2]

II.

State why the regression equations alone do not prove that distance causes longer delivery time.

[1]

Question 37

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A factory records the age, $x$ years, of several machines and the number of maintenance hours, $y$ , required in one month. The mean point is $(6,11)$ . The regression line of $y$ on $x$ has gradient $\frac{3}{2}$ , and the regression line of $x$ on $y$ has gradient $\frac{2}{5}$ .

A

I.

Find the equation of the regression line of $y$ on $x$ .

[3]

II.

Find the equation of the regression line of $x$ on $y$ .

[3]

B

I.

Use the appropriate regression line to estimate the monthly maintenance hours for a machine aged $10$ years. If you did not obtain an equation in part (a)(i), use $y=\frac{3}{2}x+2$ .

[2]

II.

Use the appropriate regression line to estimate the age of a machine requiring $15$ maintenance hours in one month. If you did not obtain an equation in part (a)(ii), use $x=\frac{2}{5}y+\frac{8}{5}$ .

[2]

C

A supervisor rearranges the regression line of $x$ on $y$ to predict $y$ from $x$ . Compare the supervisor's prediction for $x=10$ with the prediction in part (b)(i), and state which prediction should be used.

[3]

Question 38

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For a set of bivariate data, the regression line of $y$ on $x$ is

y=kx+2

and the regression line of $x$ on $y$ is

x=\frac{1}{2}y+7

The mean of the $y$ -values is $8$ . The observed values satisfy $4\leq x\leq 18$ and $3\leq y\leq 11$ .

A

I.

Find the mean of the $x$ -values.

[2]

II.

Find the value of $k$ .

[2]

B

I.

Use the appropriate regression line to estimate $y$ when $x=22$ . If you did not obtain $k=\frac{6}{11}$ , use $k=\frac{6}{11}$ .

[2]

II.

Use the appropriate regression line to estimate $x$ when $y=12$ .

[2]

C

Comment on the reliability of the two estimates in part (b), using the observed ranges.

[2]

Question 39

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A researcher records the altitude, $x$ metres, of several mountain villages and the average boiling temperature of water, $y^\circ\text{C}$ , in those villages. The regression line of $y$ on $x$ is

y=-\frac{1}{20}x+100

and the regression line of $x$ on $y$ is

x=-8y+1040

For these data, $r=-\sqrt{0.4}\approx -0.632$ . For this sample size, the critical value for testing linear correlation is $0.878$ .

A

I.

Verify that the point $(400,80)$ lies on the regression line of $y$ on $x$ .

[1]

II.

Verify that the point $(400,80)$ lies on the regression line of $x$ on $y$ .

[1]

B

I.

Use the appropriate regression line to estimate the boiling temperature at an altitude of $520$ metres.

[2]

II.

Use the appropriate regression line to estimate the altitude of a village where the average boiling temperature is $85^\circ\text{C}$ .

[2]

C

I.

Determine whether the linear correlation is significant, giving a reason.

[2]

II.

State the direction of the correlation.

[1]

D

I.

Interpret the gradient of the regression line of $x$ on $y$ in context.

[2]

II.

Explain why the regression analysis alone does not establish a causal mechanism.

[1]

Question 40

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A biologist investigates the relationship between two positive variables $x$ and $y$ . The values of $\ln x$ and $\ln y$ for six observations are
$(0.69,3.78),(1.10,3.50),(1.61,3.15),(2.08,2.87),(2.48,2.60),(2.89,2.36)$
The regression line of $\ln y$ on $\ln x$ is written as
$\ln y=a\ln x+b$

A

I.

Find the value of $a$ and the value of $b$ .

[2]

B

Assume that the relationship between $x$ and $y$ can be modelled by $y=kx^n$ , where $k>0$ .

I.

Show that $n=a$ and $k=e^b$ .

[2]

II.

Hence find the value of $n$ and the value of $k$ .

[1]

C

I.

Use the fitted regression line for $\ln y$ to estimate $y$ when $x=10$ .

[2]

II.

State whether this estimate is interpolation or extrapolation.

[1]

D

If you did not obtain a model in part (b), use $\ln y=-0.646\ln x+4.21$ . Estimate the value of $x$ when $y=25$ .

[2]

Question 41

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A researcher records values of two variables $x$ and $y$ . The paired data are
$(-3,9.1),(-2,4.2),(-1,1.0),(0,0.2),(1,1.3),(2,3.8),(3,9.4)$

A

I.

Find Pearson's product-moment correlation coefficient, $r$ , for $x$ and $y$ .

[2]

II.

Explain why this value of $r$ may be misleading if used on its own.

[2]

B

For each observation, define $z=x^2$ .

I.

Find the regression line of $y$ on $z$ .

[2]

II.

Hence estimate $y$ when $x=2.5$ .

[2]

Question 42

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

An energy company records the monthly electricity consumption, $x$ kWh, and the monthly bill, $y$ dollars, for six households. The paired data are
$(180,68),(220,75),(260,83),(310,94),(360,101),(400,112)$

A

I.

Find the regression line of $y$ on $x$ .

[2]

II.

Find the regression line of $x$ on $y$ .

[2]

B

I.

Use the appropriate regression line to estimate the electricity consumption for a household whose bill is $90$ dollars.

[2]

C

A student rearranges the regression line of $y$ on $x$ to estimate $x$ from a value of $y$ . Explain why this is not the appropriate method.

[2]

Question 43

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A drone manufacturer tests six batteries. The ambient temperature, $x^\circ\text{C}$ , and the flight time, $y$ minutes, are recorded as
$(5,31),(8,30),(12,27),(16,25),(20,22),(24,20)$

A

I.

Find the regression line of $y$ on $x$ .

[2]

II.

Find the regression line of $x$ on $y$ .

[2]

B

I.

Find the coordinates of the point of intersection of the two regression lines.

[2]

C

If you did not obtain regression lines in part (a), use $y=34.3514-0.601278x$ and $x=56.8893-1.65378y$ .

I.

Estimate the flight time when the ambient temperature is $18^\circ\text{C}$ .

[1]

II.

Estimate the ambient temperature when the flight time is $26$ minutes.

[1]

Question 44

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A conservation laboratory studies the fading of a pigment. The time, $t$ years, and the natural logarithm of the brightness index, $\ln B$ , are recorded for six samples:
$(0,4.62),(2,4.39),(4,4.15),(6,3.96),(8,3.70),(10,3.52)$
It is suggested that $B$ can be modelled by $B=Ae^{kt}$ , where $A>0$ and $k<0$ .

A

I.

Find the regression equation of $\ln B$ on $t$ in the form $\ln B=mt+c$ .

[2]

B

I.

Hence find the value of $A$ and the value of $k$ .

[3]

C

If you did not obtain a model in part (b), use $B=101e^{-0.111t}$ .

I.

Estimate the time at which the brightness index is $50$ .

[2]

D

I.

State whether the estimate in part (c) is interpolation or extrapolation.

[1]

II.

Give one limitation of this model.

[1]

Question 45

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A jeweller records the diameter, $x$ mm, and mass, $y$ mg, of six gemstones:
$(4,18),(5,34),(6,25),(7,58),(8,49),(9,84)$

A

I.

Find the regression line of $y$ on $x$ .

[2]

II.

Find the regression line of $x$ on $y$ .

[2]

B

If you did not obtain regression equations in part (a), use $y=11.657143x-31.104762$ and $x=0.0690y+3.42$ for subsequent calculations.

I.

Estimate the mass of a gemstone with diameter $7.5$ mm.

[1]

II.

Estimate the diameter of a gemstone with mass $50$ mg.

[1]

C

A student uses the equation $y=11.7x-31.1$ to estimate the diameter of a gemstone with mass $50$ mg. Compare this method with the method used in part (b)(ii).

[2]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A manufacturer tests rechargeable cells at different ambient temperatures. Let $x$ be the ambient temperature in $^\circ\text{C}$ and let $y$ be the discharge time in hours. For a sample of $9$ cells, the following summary statistics are obtained:

\bar{x}=25,\quad \bar{y}=6.80,\quad s_x=4.60,\quad s_y=1.15,\quad r=-0.872

A scatter diagram suggests that a linear model is reasonable for temperatures between $18^\circ\text{C}$ and $32^\circ\text{C}$ .

A

I.

Find the equation of the regression line of $y$ on $x$ in the form $y=ax+b$ .

[3]

II.

Use your regression line to estimate the discharge time at $22^\circ\text{C}$ .

[2]

B

I.

Find the equation of the regression line of $x$ on $y$ in the form $x=cy+d$ .

[3]

II.

Estimate the ambient temperature corresponding to a discharge time of $5.80$ hours. If you did not obtain an equation in part (b)(i), use $x=-3.49y+48.7$ .

[2]

C

I.

Show that the product of the gradients of the two regression lines is equal to $r^2$ .

[2]

II.

Explain why rearranging the regression line of $y$ on $x$ is not the correct method for estimating $x$ from a given value of $y$ .

[2]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

An environmental officer records the level of fine particles, $x$ , and the number of respiratory complaints, $y$ , in $8$ districts. One district is known to contain a large industrial fire during the recording period. For all $8$ districts,

\bar{x}=42, \quad \bar{y}=18, \quad S_{xx}=1246, \quad S_{yy}=1018.5, \quad S_{xy}=1099

where $S_{xx}=\sum (x-\bar{x})^2$ , $S_{yy}=\sum (y-\bar{y})^2$ and $S_{xy}=\sum (x-\bar{x})(y-\bar{y})$ .

After removing the industrial-fire district, the corresponding statistics for the remaining $7$ districts are

\bar{x}=38, \quad \bar{y}=14, \quad S_{xx}=350, \quad S_{yy}=122.5, \quad S_{xy}=203

Statistic	All 8 districts	Remaining 7 districts
$n$	8	7
Mean $x$	42	38
Mean $y$	18	14
$S_{xx}$	1246	350
$S_{yy}$	1018.5	122.5
$S_{xy}$	1099	203

A

I.

Calculate Pearson's product-moment correlation coefficient for all $8$ districts.

[2]

II.

Find the regression line of $y$ on $x$ using all $8$ districts.

[3]

B

I.

Calculate Pearson's product-moment correlation coefficient after removing the industrial-fire district.

[2]

II.

Using the remaining $7$ districts, find the regression line of $y$ on $x$ .

[2]

C

I.

Using the model from part (b)(ii), estimate the number of complaints when the particle level is $45$ . If you did not obtain an equation in part (b)(ii), use $y=0.580x-8.04$ .

[2]

II.

The critical values of $|r|$ for a two-tailed test at the $5\%$ level are $0.707$ for $n=8$ and $0.754$ for $n=7$ . Compare the conclusions about significant linear correlation before and after removing the industrial-fire district.

[2]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A materials laboratory measures the diameter $x$ mm and breaking load $y$ N of metal rods. Technology gives the regression line of $y$ on $x$ as

y=18.5x+33.0

The regression line of $x$ on $y$ is

x=0.0410y+1.545

The observed loads range from $190$ N to $335$ N.

Use the stated regression equations, rather than numerical values read from the diagram, for all calculations.

A

I.

Find the coordinates of the mean point of the data.

[3]

II.

Find Pearson's product-moment correlation coefficient, $r$ .

[1]

B

I.

Estimate the diameter of a rod with breaking load $310$ N. State which regression line you used.

[2]

II.

A student instead rearranges $y=18.5x+33.0$ to estimate the diameter. Find the student's estimate.

[2]

C

I.

Explain why the estimate in part (b)(i) is more appropriate than the student's estimate.

[2]

II.

Deduce why the two regression lines would be rearrangements of each other only when $|r|=1$ .

[2]

Question 49

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A chemical engineer records the temperature $x$ in $^\circ\text{C}$ and the time $y$ in minutes for a reaction to reach a fixed yield. Technology gives

y=-0.0450x+6.20

as the regression line of $y$ on $x$ , and

x=-14.2y+103

as the regression line of $x$ on $y$ .

The engineer later uses $u=x+273.15$ for temperature in kelvin and $v=60y$ for time in seconds.

A

I.

Find Pearson's product-moment correlation coefficient between $x$ and $y$ .

[2]

II.

Explain why the negative value of $r$ is appropriate in this context.

[2]

B

I.

Find the regression line of $v$ on $u$ .

[3]

II.

Use your line to estimate the reaction time in seconds at $295$ K. If you did not obtain an equation in part (b)(i), use $v=-2.70u+1109.505$ , retaining unrounded values before final rounding.

[2]

C

I.

Find the regression line of $u$ on $v$ .

[2]

II.

State the correlation coefficient between $u$ and $v$ , giving a reason.

[2]

Question 50

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A biologist studies the relationship between salinity $x$ and the number of larvae $y$ found in water samples. Technology gives the two regression lines

y=-4.20x+58.6

and

x=-0.207y+13.14

There are $11$ samples and the observed salinities range from $2.1$ to $8.9$ .

A

I.

Find Pearson's product-moment correlation coefficient, $r$ .

[2]

II.

Describe the linear correlation.

[1]

B

I.

Estimate the number of larvae when the salinity is $6.00$ . State whether this is interpolation or extrapolation.

[2]

II.

Estimate the salinity when the number of larvae is $25$ . State which regression line is used.

[2]

C

I.

For $n=11$ , the critical value of $|r|$ at the $5\%$ significance level is $0.602$ . Determine whether there is significant linear correlation.

[2]

II.

The biologist claims that increasing salinity causes the number of larvae to decrease. Comment on this claim.

[2]

III.

Explain why using the first regression line to estimate $y$ when $x=11$ may be unreliable.

[1]

Question 51

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A workplace study records background noise level $x$ dB and productivity score $y$ for $10$ offices. Technology gives

r=-0.646, \quad y=-0.580x+91.4

For $n=10$ , the critical values of $|r|$ for a two-tailed test are $0.632$ at the $5\%$ level and $0.765$ at the $1\%$ level. A new variable $u=x-60$ measures noise above $60$ dB, and $v=100-y$ measures productivity loss.

A

I.

State suitable hypotheses for testing whether there is linear correlation between noise level and productivity score.

[1]

II.

Determine whether the correlation is significant at the $5\%$ level and at the $1\%$ level.

[3]

B

I.

Interpret the gradient of the regression line in context.

[2]

II.

Explain why the intercept should be interpreted with caution.

[2]

C

I.

Find the regression line of $v$ on $u$ .

[2]

II.

State the correlation coefficient between $u$ and $v$ , giving a reason.

[2]

Question 52

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For a set of bivariate data, the regression line of $y$ on $x$ is

y=-\frac{1}{2}x+18

and the regression line of $x$ on $y$ is

x=-\frac{1}{8}y+9

A

Find the mean point of the data.

[4]

B

I.

Use the appropriate regression line to estimate $y$ when $x=6$ .

[2]

II.

Use the appropriate regression line to estimate $x$ when $y=10$ .

[2]

C

I.

Show algebraically that the two regression lines would be the same line only if their gradients were reciprocals.

[3]

II.

Use the gradients in this question to explain why the two regression lines are not the same line.

[1]

Question 53

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A small wind turbine is tested at different wind speeds. Let $v$ be the wind speed and $P$ be the power output. The following paired values of $\ln v$ and $\ln P$ are recorded:
$(1.10,3.02),(1.39,3.86),(1.61,4.55),(1.79,5.07),(1.95,5.52),(2.08,5.93)$

A

I.

Find the regression line of $\ln P$ on $\ln v$ in the form $\ln P=a \ln v+b$ .

[2]

II.

Hence write a model for $P$ in the form $P=kv^n$ .

[2]

B

I.

Find the regression line of $\ln v$ on $\ln P$ .

[2]

II.

Use the appropriate regression equation to estimate the wind speed when the power output is $150$ . If you did not obtain an equation in part (b)(i), use $\ln v=0.337\ln P+0.0832$ .

[2]

C

Explain why rearranging the equation in part (a)(i) is not generally the appropriate method for estimating $v$ from a given value of $P$ .

[2]

Question 54

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A property analyst records the floor number, $x$ , and the weekly rent, $y$ dollars, for seven apartments in the same building:
$(1,210),(2,220),(3,235),(4,245),(5,260),(6,270),(7,410)$
The apartment on floor $7$ has a large private terrace and may be an unusual observation.

A

I.

Using all seven observations, find Pearson's product-moment correlation coefficient, $r$ .

[2]

II.

Remove the observation for floor $7$ . Find the new value of $r$ for the remaining six observations.

[2]

B

I.

Find the regression line of $y$ on $x$ using all seven observations.

[1]

II.

Find the regression line of $y$ on $x$ using only the first six observations.

[1]

III.

Use both regression lines to estimate the weekly rent on floor $8$ .

[1]

C

Evaluate the reliability of using these models to predict the rent on floor $8$ .

[2]

Question 55

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A data set has six observations for variables $x$ and $y$ :
$(10,44),(15,52),(20,61),(25,69),(30,80),(35,87)$
New variables $u$ and $v$ are defined by
$u=\frac{x-20}{5},\qquad v=2y+15$

$x$	$y$	$u$	$v$
10	44	-2	103
15	52	-1	119
20	61	0	137
25	69	1	153
30	80	2	175
35	87	3	189

A

I.

Find Pearson's product-moment correlation coefficient for $x$ and $y$ .

[2]

II.

Find the regression line of $y$ on $x$ .

[2]

B

I.

Determine Pearson's product-moment correlation coefficient for $u$ and $v$ .

[1]

II.

Find the regression line of $v$ on $u$ .

[2]

C

If you did not obtain an equation in part (b)(ii), use $v=17.5u+137$ . Estimate $v$ when $u=3$ .

[1]

Question 56

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A light sensor is moved to positions at signed distances $x$ cm from the centre of a lamp. The recorded intensity is $y$ . The scatter diagram is symmetric about the $y$ -axis. For the data collected, the transformation $z=x^2$ gives an exact linear relationship

y=z+1

The observed values of $x$ are equally spaced on both sides of $0$ , and no observation is made at $x=0$ .

x [cm]	y [a.u.]	z [cm^2]
-3	10	9
-2	5	4
-1	2	1
1	2	1
2	5	4
3	10	9

A

I.

Explain why the Pearson correlation coefficient between $x$ and $y$ is $0$ .

[2]

II.

Explain why $r=0$ does not mean that there is no relationship between $x$ and $y$ .

[2]

B

I.

Use the regression of $y$ on $z$ to estimate the intensity when $x=2.50$ .

[2]

II.

State whether this estimate is an interpolation or an extrapolation, given that $2.50$ lies within the observed range of $x$ .

[2]

C

I.

Suppose a data set consists of pairs $(a_i,f(a_i))$ and $(-a_i,f(a_i))$ , where $a_i>0$ . Show that the Pearson correlation coefficient between $x$ and $y$ is $0$ , provided the variance of $x$ and the variance of $y$ are non-zero.

[4]

Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A meteorologist studies the relationship between wind speed $x$ km h $^{-1}$ and a cooling index $y$ . For the sample,

\bar{x}=30, \quad \bar{y}=12, \quad S_{xx}=160, \quad S_{xy}=-96, \quad S_{yy}=74

The plotted observations are illustrative and rounded; use the supplied summary statistics for all calculations.

A family of straight lines through the mean point is considered:

y-12=k(x-30)

A

I.

Find Pearson's product-moment correlation coefficient, $r$ .

[2]

II.

Find the regression line of $y$ on $x$ .

[2]

B

I.

Show that the sum of squared vertical residuals for the line $y-12=k(x-30)$ is

Q(k)=S_{yy}-2kS_{xy}+k^2S_{xx}

[3]

II.

Hence find the value of $k$ that minimizes $Q(k)$ .

[2]

C

I.

Find the minimum possible value of $Q(k)$ . If you did not obtain $k=-0.600$ , use this value.

[2]

II.

For an observation with $x=35$ and $y=8.10$ , find the residual using the regression line of $y$ on $x$ .

[2]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A geologist records the mineral density $x$ and magnetic response $y$ of rock samples. For one analysis, the regression line of $x$ on $y$ is

x=0.240y+1.80

and the mean point is $(9.00,30.0)$ . The Pearson correlation coefficient is $r=0.600$ .

A

I.

Verify that the given regression line passes through the mean point.

[1]

II.

Find the equation of the regression line of $y$ on $x$ .

[3]

B

I.

A second report claims that the regression line of $y$ on $x$ is $y=5.00x-15.0$ while the regression line of $x$ on $y$ remains $x=0.240y+1.80$ . Explain why this cannot be correct.

[2]

II.

Use the correct regression line of $y$ on $x$ to estimate $y$ when $x=12.0$ . If you did not obtain an equation in part (a)(ii), use $y=1.50x+16.5$ .

[1]

C

I.

Deduce why, unless $|r|=1$ , the regression line of $x$ on $y$ should not be rearranged to predict $y$ from a given $x$ .

[3]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A researcher studies the relationship between the number of practice problems attempted, $x$ , and a performance score, $y$ . For the first $6$ students,

\bar{x}=10, \quad \bar{y}=40, \quad S_{xx}=70, \quad S_{yy}=280, \quad S_{xy}=126

A seventh student, who attempted many practice problems but had a relatively low score, is added to the data set. This student's values are $(20,42)$ .

A

I.

For the first $6$ students, find Pearson's correlation coefficient.

[2]

II.

Find the regression line of $y$ on $x$ for the first $6$ students.

[3]

B

I.

Find the new mean point after the seventh student is added.

[2]

II.

Show that the updated values are

S_{xx}=\frac{1090}{7},\quad S_{yy}=\frac{1984}{7},\quad S_{xy}=\frac{1002}{7}

[3]

C

I.

For all $7$ students, find the correlation coefficient and the regression line of $y$ on $x$ . If you did not show the updated values in part (b)(ii), use them as given.

[3]

II.

Compare the predicted score for $x=15$ using the original model and the updated model.

[2]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A nutrition study records daily fibre intake $x$ grams and a health score $y$ . The participants belong to two age groups, A and B. For group A,

n=5, \quad \bar{x}=2, \quad \bar{y}=18, \quad S_{xx}=10, \quad S_{yy}=40, \quad S_{xy}=20

For group B,

n=5, \quad \bar{x}=8, \quad \bar{y}=12, \quad S_{xx}=10, \quad S_{yy}=40, \quad S_{xy}=20

A

I.

Find the correlation coefficient within each group.

[2]

II.

Find the regression line of $y$ on $x$ within group A.

[2]

B

I.

Find the combined mean point for all $10$ participants.

[2]

II.

Show that, for the combined data, $S_{xx}=110$ , $S_{yy}=170$ and $S_{xy}=-50$ .

[4]

C

I.

Find the correlation coefficient and the regression line of $y$ on $x$ for the combined data.

[2]

II.

Discuss why the combined correlation may lead to a misleading interpretation of the relationship between fibre intake and health score.

[2]